Theorem: Combined capture certificate

For the baseline participating marginal and every admitted normal spectator–reference extension, the capture-plus-storage channel obeys

Proof. Replace the independent memory preparation by vacuum at cost $p_c$. By (eq:capturepair), the resulting reduced capture and storage are joint vacuum pure loss on the orthonormal $F$-pair. Lemma lem:jointloss costs $\ell (E_F,{\tau }_{\ast })$. Finally compare the $F$-selector and target selector using Lemma lem:mode. All comparisons retain $SR$, and contractivity gives the result.